Airfoil Shape for a Turbine Rotor Blade

ABSTRACT

A turbine rotor blade is provided including an airfoil having an airfoil shape. The airfoil shape having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z, the airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape.

FIELD

The present invention relates to an airfoil for a turbine rotor blade of a stage of a turbine and particularly relates to a shape defining a profile for an airfoil of a turbine rotor blade.

BACKGROUND

Some aircraft and/or power plant systems, for example certain jet aircraft, nuclear, simple cycle and combined cycle power plant systems, employ turbines (also referred to as turbomachines) in their design and operation. Some of these turbines employ airfoils (e.g., stator vanes or nozzles and rotor blades) which during operation are exposed to fluid flows. These airfoils are configured to aerodynamically interact with the fluid flows and generate energy (e.g., creating thrust, turning kinetic energy to mechanical energy, thermal energy to mechanical energy, etc.) from these fluid flows as part of power generation. As a result of this interaction and conversion, the aerodynamic characteristics and losses of these airfoils have an impact on system and turbine operation, performance, thrust, efficiency, and power.

BRIEF DESCRIPTION

Aspects and advantages are set forth below in the following description, or may be obvious from the description, or may be learned through practice.

An aspect of the present invention may be embodied by a turbine rotor blade including an airfoil having an airfoil shape, the airfoil having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z, the airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape.

An aspect of the present invention may be embodied in a turbine rotor blade including an airfoil having a suction-side uncoated nominal airfoil profile substantially in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at the Z distances being joined smoothly with one another to form a complete suction-side airfoil shape, the X, Y and Z distances being scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil.

An aspect of the present invention may be embodied in a turbine comprising a turbine wheel having a plurality of turbine rotor blades, each of the turbine rotor blades including an airfoil having a suction-side airfoil shape, the airfoil having a nominal profile substantially in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil, and wherein X and Y are distances which, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at the Z distances being joined smoothly with one another to form a complete suction-side airfoil shape.

Those of ordinary skill in the art will better appreciate the features and aspects of such embodiments, and others, upon review of the specification.

BRIEF DESCRIPTION OF THE DRAWINGS

A full and enabling disclosure of various embodiments, including the best mode thereof to one skilled in the art, is set forth more particularly in the remainder of the specification, including reference to the accompanying figures, in which:

FIG. 1 is a schematic representation of an exemplary turbine having multiple stages with each stage including alternating rows of turbine rotor blades and stationary vanes or nozzles according to at least one embodiment of the present disclosure; and

FIG. 2 is a perspective view of a turbine rotor blade according to at least one embodiment of the present disclosure.

DETAILED DESCRIPTION

Referring now to the drawings, particularly to FIG. 1, there is illustrated an exemplary turbine 10 of a gas turbine 12 including a plurality of turbine stages arranged in serial flow order. Each stage of the turbine includes a row of turbine nozzles or vanes disposed axially adjacent to a corresponding row of turbine rotor blades. Four stages are illustrated in FIG. 1. The exact number of stages of the turbine 10 is a choice of engineering design, and may be more or less than the four stages illustrated in FIG. 1. The four stages are merely exemplary of one turbine design, and are not intended to limit the presently claimed turbine rotor blade in any manner.

The first stage comprises a plurality of turbine nozzles or vanes 100 and a plurality of turbine rotor blades 150. The turbine nozzles 100 are annularly arranged about an axis of a turbine rotor 14. The turbine rotor blades 150 are annularly arranged about and coupled to the turbine rotor 14. A second stage of the turbine 12 includes a plurality of turbine nozzles 200 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 250 annularly arranged about and coupled to the turbine rotor 14. The turbine nozzles 200 may be coupled to a casing 16 of the turbine 12.

As shown in FIG. 1, a third stage of the turbine 12 includes a plurality of turbine nozzles 300 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 350 annularly arranged about and coupled to the turbine rotor 14. The turbine nozzles 300 may be coupled to the casing 16 of the turbine 12. In particular embodiments, the turbine 12 includes a fourth stage which includes plurality of turbine nozzles 400 annularly arranged about the axis of the turbine rotor 14 and a plurality of turbine rotor blades 450 annularly arranged about and coupled to the turbine rotor 14. The fourth stage turbine nozzles 400 may be coupled to the casing 16 of the turbine 12.

It will be appreciated that the turbine nozzles 100, 200, 300 and 400 and turbine rotor blades 150, 250, 350 and 450 are disposed or at least partially disposed within a hot gas path 18 of the turbine 12. The various stages of the turbine 10 at least partially define the hot gas path 18 through which combustion gases, as indicated by arrows 20, flow during operation of the gas turbine 12.

FIG. 2 provides a perspective view of an exemplary turbine rotor blade 50 as may be incorporated in any stage of the turbine 12. For example, in one embodiment the turbine rotor blade 50 as presented in Table I below is representative of a turbine rotor blade 350 of the plurality of turbine rotor blades 350 of the third stage. As shown in FIG. 2, the turbine rotor blade 50 includes an airfoil 52 having a pressure-side wall 54 and an opposing suction-side wall 56. The pressure-side wall 54 and the suction-side wall 56 meet or intersect at a leading edge 58 and a trailing edge 60 of the airfoil 52. A chord line 62 extends between the leading edge 58 and the trailing edge 60 such that pressure and suction-side walls 54, 56 can be said to extend in chord or chordwise between the leading edge 58 and the trailing edge 60.

The airfoil 52 includes a root or first end 64 which intersects with and extends radially outwardly from a base or platform 66 of the turbine rotor blade 50. The airfoil 52 terminates radially at a second end or radial tip 68 of the airfoil 52. The pressure and suction-side walls 54, 56 can be said to extend in span or in a span-wise direction 70 between the root 64 and/or the platform 66 and the radial tip 68 of the airfoil 52. In other words, each turbine rotor blade 50 includes an airfoil 52 having opposing pressure and suction-side walls 54, 56 that extend in chord or chordwise 62 between opposing leading and trailing edges 58, 60 and that extend in span or span-wise 70 between the root 64 and the radial tip 68 of the airfoil 52.

In particular configurations, the airfoil 52 may include a fillet 72 formed between the platform 66 and the airfoil 52 proximate to the root 64. The fillet 72 can include a weld or braze fillet, which can be formed via conventional MIG welding, TIG welding, brazing, etc., and can include a profile that can reduce fluid dynamic losses as a result of the presence of fillet 72. In particular embodiments, the platform 66, airfoil 52 and the fillet 72 can be formed as a single component, such as by casting and/or machining and/or 3D printing and/or any other suitable technique now known or later developed and/or discovered. In particular configurations, the turbine rotor blade 50 includes a mounting portion 74 which is formed to connect and/or to secure the turbine rotor blade 50 to the rotor shaft 14.

The airfoil 52 of the turbine rotor blade 50 has a profile at any cross-section taken between the platform 66 or the root 64 and the radial tip 68. The X, Y, and Z values are given in Table I as percentage values of the airfoil length or height/span. As one example only, the height or radial span of the airfoil 52 of turbine rotor blade 50 may be from about 10 inches to about 40 inches. However, it is to be understood that heights below or above this range may also be employed as desired in the specific application.

A gas turbine hot gas path requires airfoils that meet system requirements of aerodynamic and mechanical blade loading and efficiency. To define the airfoil shape of each turbine rotor blade airfoil, there is a unique set or loci of points in space that meet the stage requirements and that can be manufactured. This unique loci of points meets the requirements for stage efficiency and are arrived at by iteration between aerodynamic and mechanical loadings enabling the turbine to run in an efficient, safe and smooth manner. These points are unique and specific to the system and are not obvious to those skilled in the art.

The loci which defines the airfoil profile of turbine rotor blade 50 comprises a set of about 2,200 points with X, Y and Z dimensions relative to a reference origin coordinate system. The Cartesian coordinate system of X, Y and Z values given in Table I below defines the airfoil profile of the turbine rotor blade 50 at various locations along its length and height or radial span. Table I lists data for a non-coated airfoil, and the envelope/tolerance for the coordinates is about +/−5% in a direction normal to any airfoil surface location. A point data origin 76 is defined at or proximate to the root 64 at the leading edge 58. In one embodiment as presented in Table I below, the point data origin 76 is defined at or proximate to (above or below) a transition or intersection line 78 defined between the fillet 72 and the airfoil 52. The point data origin 76 corresponds to the non-dimensional Z value of Table I (presented below) at Z equals 0.

The coordinate values for the X, Y and Z coordinates are set forth in non-dimensionalized units by the airfoil height in Table I although other units of dimensions may be used when the values are appropriately converted. The X, Y, and Z values set forth in Table I are also expressed in non-dimensional form (X, Y, and Z) from 0% to 100% of the airfoil 52 height or radial span 70. To convert the Z value to a Z coordinate value, e.g., in inches, the non-dimensional Z value given in Table I is multiplied by the Z length of the airfoil in inches. As described above, the Cartesian coordinate system has orthogonally-related X, Y and Z axes and the X axis lies generally parallel to a centerline of the rotor shaft 14, i.e., the rotary axis and a positive X coordinate value is axial toward an aft, i.e., exhaust end of the turbine 10. The positive Y coordinate value extends tangentially in the direction of rotation of the rotor and the positive Z coordinate value is radially outwardly toward the radial tip 68. All the values in Table I are given at room temperature and do not include the fillet 72.

By defining X and Y coordinate values at selected locations in a Z direction normal to the X, Y plane, the profile section or airfoil shape of the airfoil 52 of the turbine rotor blade 50, at each Z distance along the length of the airfoil 52 can be ascertained. By connecting the X and Y values with smooth continuing arcs, each profile section at each distance Z is fixed. The airfoil profiles of the various surface locations between the distances Z are determined by smoothly connecting the adjacent profile sections to one another to form the airfoil profile.

The Table I values are generated and shown to four decimal places for determining the profile of the airfoil 52. As the turbine rotor blade 50 heats up during operation of the gas turbine 12, surface stress and temperature will cause a change in the X, Y and Z's. Accordingly, the values for the profile given in Table I represent ambient, non-operating or non-hot conditions (e.g., room temperature) and are for an uncoated airfoil.

There are typical manufacturing tolerances as well as coatings which must be accounted for in the actual profile of the airfoil 52. Each cross-section is joined smoothly with the other cross-sections to form the complete airfoil shape. It will therefore be appreciated that +/− typical manufacturing tolerances, i.e., +/− values, including any coating thicknesses, are additive to the X and Y values given in Table I below. Accordingly, a distance of +/−5% in a direction normal to any surface location along the airfoil profile defines an airfoil profile envelope for this particular turbine rotor blade 50 airfoil design and turbine 12, i.e., a range of variation between measured points on the actual airfoil surface at nominal cold or room temperature and the ideal position of those points as given in Table I below at the same temperature. The data provided in Table I is scalable and the geometry pertains to all aerodynamic scales, at, above and/or below a 3000-3600 RPM range. The design of the airfoil 52 for turbine rotor blade 50 is robust to this range of variation without impairment of mechanical and aerodynamic functions.

TABLE I SUCTION-SIDE PRESSURE-SIDE X (%) Y (%) Z (%) N X (%) Y (%) Z (%) 0.0000 0.0000 0.0000 1 0.0000 0.0000 0.0000 −0.1741 0.3908 0.0000 2 0.2870 −0.1968 0.0000 −0.1254 0.8211 0.0000 3 0.6195 −0.3086 0.0000 0.0400 1.2227 0.0000 4 0.9649 −0.3670 0.0000 0.2676 1.5941 0.0000 5 1.3146 −0.3908 0.0000 0.5324 1.9395 0.0000 6 1.6654 −0.3886 0.0000 0.8232 2.2632 0.0000 7 2.0157 −0.3670 0.0000 1.1341 2.5681 0.0000 8 2.3643 −0.3292 0.0000 1.4584 2.8589 0.0000 9 2.7114 −0.2795 0.0000 1.7924 3.1384 0.0000 10 3.0578 −0.2222 0.0000 2.1357 3.4070 0.0000 11 3.4032 −0.1611 0.0000 2.4849 3.6665 0.0000 12 3.7481 −0.0989 0.0000 2.8405 3.9178 0.0000 13 4.0935 −0.0378 0.0000 3.2016 4.1622 0.0000 14 4.4395 0.0216 0.0000 3.5665 4.3995 0.0000 15 4.7854 0.0784 0.0000 3.9357 4.6303 0.0000 16 5.1319 0.1335 0.0000 4.3086 4.8557 0.0000 17 5.4789 0.1859 0.0000 4.6849 5.0751 0.0000 18 5.8259 0.2368 0.0000 5.0643 5.2881 0.0000 19 6.1735 0.2849 0.0000 5.4470 5.4962 0.0000 20 6.5211 0.3314 0.0000 5.8330 5.6978 0.0000 21 6.8692 0.3757 0.0000 6.2222 5.8935 0.0000 22 7.2173 0.4184 0.0000 6.6141 6.0832 0.0000 23 7.5654 0.4584 0.0000 7.0092 6.2665 0.0000 24 7.9141 0.4968 0.0000 7.4076 6.4427 0.0000 25 8.2632 0.5330 0.0000 7.8086 6.6119 0.0000 26 8.6124 0.5670 0.0000 8.2130 6.7735 0.0000 27 8.9616 0.5984 0.0000 8.6205 6.9276 0.0000 28 9.3114 0.6276 0.0000 9.0308 7.0735 0.0000 29 9.6611 0.6546 0.0000 9.4438 7.2114 0.0000 30 10.0108 0.6784 0.0000 9.8600 7.3400 0.0000 31 10.3611 0.7000 0.0000 10.2784 7.4600 0.0000 32 10.7114 0.7189 0.0000 10.7000 7.5703 0.0000 33 11.0616 0.7346 0.0000 11.1238 7.6708 0.0000 34 11.4124 0.7476 0.0000 11.5497 7.7611 0.0000 35 11.7627 0.7573 0.0000 11.9778 7.8411 0.0000 36 12.1135 0.7643 0.0000 12.4076 7.9103 0.0000 37 12.4643 0.7681 0.0000 12.8395 7.9681 0.0000 38 12.8151 0.7681 0.0000 13.2724 8.0146 0.0000 39 13.1659 0.7654 0.0000 13.7065 8.0486 0.0000 40 13.5168 0.7589 0.0000 14.1416 8.0708 0.0000 41 13.8676 0.7486 0.0000 14.5768 8.0800 0.0000 42 14.2178 0.7351 0.0000 15.0124 8.0762 0.0000 43 14.5681 0.7178 0.0000 15.4476 8.0600 0.0000 44 14.9184 0.6968 0.0000 15.8822 8.0297 0.0000 45 15.2681 0.6714 0.0000 16.3151 7.9859 0.0000 46 15.6178 0.6427 0.0000 16.7470 7.9286 0.0000 47 15.9670 0.6092 0.0000 17.1768 7.8573 0.0000 48 16.3157 0.5719 0.0000 17.6038 7.7719 0.0000 49 16.6638 0.5303 0.0000 18.0276 7.6724 0.0000 50 17.0119 0.4838 0.0000 18.4481 7.5589 0.0000 51 17.3589 0.4335 0.0000 18.8643 7.4319 0.0000 52 17.7054 0.3778 0.0000 19.2768 7.2908 0.0000 53 18.0508 0.3178 0.0000 19.6832 7.1357 0.0000 54 18.3957 0.2530 0.0000 20.0849 6.9670 0.0000 55 18.7395 0.1827 0.0000 20.4805 6.7854 0.0000 56 19.0822 0.1081 0.0000 20.8697 6.5897 0.0000 57 19.4232 0.0281 0.0000 21.2524 6.3822 0.0000 58 19.7638 −0.0573 0.0000 21.6281 6.1611 0.0000 59 20.1027 −0.1476 0.0000 21.9962 5.9286 0.0000 60 20.4400 −0.2432 0.0000 22.3568 5.6843 0.0000 61 20.7762 −0.3443 0.0000 22.7092 5.4286 0.0000 62 21.1103 −0.4514 0.0000 23.0535 5.1622 0.0000 63 21.4427 −0.5632 0.0000 23.3903 4.8854 0.0000 64 21.7730 −0.6811 0.0000 23.7184 4.5995 0.0000 65 22.1016 −0.8043 0.0000 24.0389 4.3043 0.0000 66 22.4276 −0.9330 0.0000 24.3514 4.0011 0.0000 67 22.7514 −1.0681 0.0000 24.6562 3.6903 0.0000 68 23.0730 −1.2086 0.0000 24.9535 3.3724 0.0000 69 23.3914 −1.3551 0.0000 25.2438 3.0476 0.0000 70 23.7076 −1.5070 0.0000 25.5270 2.7168 0.0000 71 24.0205 −1.6654 0.0000 25.8038 2.3805 0.0000 72 24.3308 −1.8292 0.0000 26.0741 2.0389 0.0000 73 24.6378 −1.9989 0.0000 26.3384 1.6930 0.0000 74 24.9416 −2.1741 0.0000 26.5973 1.3427 0.0000 75 25.2422 −2.3551 0.0000 26.8508 0.9886 0.0000 76 25.5395 −2.5411 0.0000 27.0995 0.6308 0.0000 77 25.8335 −2.7330 0.0000 27.3432 0.2697 0.0000 78 26.1238 −2.9297 0.0000 27.5827 −0.0941 0.0000 79 26.4097 −3.1330 0.0000 27.8178 −0.4600 0.0000 80 26.6914 −3.3416 0.0000 28.0497 −0.8292 0.0000 81 26.9686 −3.5568 0.0000 28.2778 −1.2000 0.0000 82 27.2405 −3.7778 0.0000 28.5032 −1.5730 0.0000 83 27.5076 −4.0054 0.0000 28.7254 −1.9470 0.0000 84 27.7692 −4.2395 0.0000 28.9449 −2.3232 0.0000 85 28.0249 −4.4795 0.0000 29.1622 −2.7005 0.0000 86 28.2751 −4.7249 0.0000 29.3773 −3.0795 0.0000 87 28.5205 −4.9762 0.0000 29.5903 −3.4595 0.0000 88 28.7600 −5.2324 0.0000 29.8011 −3.8400 0.0000 89 28.9946 −5.4930 0.0000 30.0103 −4.2227 0.0000 90 29.2238 −5.7584 0.0000 30.2168 −4.6059 0.0000 91 29.4481 −6.0281 0.0000 30.4211 −4.9903 0.0000 92 29.6676 −6.3016 0.0000 30.6227 −5.3762 0.0000 93 29.8827 −6.5789 0.0000 30.8216 −5.7638 0.0000 94 30.0924 −6.8600 0.0000 31.0173 −6.1530 0.0000 95 30.2984 −7.1443 0.0000 31.2097 −6.5438 0.0000 96 30.5000 −7.4314 0.0000 31.3984 −6.9362 0.0000 97 30.6978 −7.7211 0.0000 31.5832 −7.3303 0.0000 98 30.8924 −8.0124 0.0000 31.7205 −7.7395 0.0000 99 31.1854 −8.1886 0.0000 31.5184 −8.1059 0.0000 100 31.5184 −8.1059 0.0000 0.9335 0.8270 10.0000 1 0.9335 0.8270 10.0000 0.8205 1.2238 10.0000 2 1.1957 0.6103 10.0000 0.9189 1.6308 10.0000 3 1.5157 0.4897 10.0000 1.1146 2.0027 10.0000 4 1.8530 0.4276 10.0000 1.3573 2.3465 10.0000 5 2.1946 0.4027 10.0000 1.6292 2.6676 10.0000 6 2.5378 0.4032 10.0000 1.9211 2.9708 10.0000 7 2.8800 0.4243 10.0000 2.2270 3.2595 10.0000 8 3.2205 0.4627 10.0000 2.5438 3.5362 10.0000 9 3.5600 0.5130 10.0000 2.8697 3.8022 10.0000 10 3.8978 0.5708 10.0000 3.2038 4.0589 10.0000 11 4.2357 0.6314 10.0000 3.5438 4.3065 10.0000 12 4.5730 0.6924 10.0000 3.8892 4.5465 10.0000 13 4.9108 0.7524 10.0000 4.2400 4.7789 10.0000 14 5.2486 0.8097 10.0000 4.5951 5.0049 10.0000 15 5.5870 0.8643 10.0000 4.9546 5.2238 10.0000 16 5.9265 0.9157 10.0000 5.3173 5.4368 10.0000 17 6.2659 0.9643 10.0000 5.6843 5.6438 10.0000 18 6.6054 1.0108 10.0000 6.0541 5.8443 10.0000 19 6.9454 1.0546 10.0000 6.4270 6.0389 10.0000 20 7.2859 1.0962 10.0000 6.8038 6.2265 10.0000 21 7.6265 1.1357 10.0000 7.1838 6.4076 10.0000 22 7.9676 1.1724 10.0000 7.5670 6.5816 10.0000 23 8.3086 1.2070 10.0000 7.9535 6.7481 10.0000 24 8.6503 1.2384 10.0000 8.3432 6.9070 10.0000 25 8.9919 1.2676 10.0000 8.7362 7.0578 10.0000 26 9.3341 1.2930 10.0000 9.1319 7.2005 10.0000 27 9.6762 1.3157 10.0000 9.5308 7.3346 10.0000 28 10.0184 1.3346 10.0000 9.9330 7.4595 10.0000 29 10.3611 1.3508 10.0000 10.3373 7.5751 10.0000 30 10.7038 1.3638 10.0000 10.7449 7.6805 10.0000 31 11.0465 1.3730 10.0000 11.1546 7.7762 10.0000 32 11.3897 1.3795 10.0000 11.5665 7.8611 10.0000 33 11.7324 1.3827 10.0000 11.9811 7.9351 10.0000 34 12.0757 1.3822 10.0000 12.3973 7.9978 10.0000 35 12.4184 1.3778 10.0000 12.8146 8.0486 10.0000 36 12.7611 1.3692 10.0000 13.2341 8.0876 10.0000 37 13.1038 1.3568 10.0000 13.6541 8.1141 10.0000 38 13.4465 1.3405 10.0000 14.0746 8.1276 10.0000 39 13.7886 1.3200 10.0000 14.4951 8.1276 10.0000 40 14.1308 1.2957 10.0000 14.9157 8.1146 10.0000 41 14.4724 1.2670 10.0000 15.3357 8.0881 10.0000 42 14.8141 1.2341 10.0000 15.7546 8.0476 10.0000 43 15.1546 1.1968 10.0000 16.1719 7.9935 10.0000 44 15.4951 1.1546 10.0000 16.5870 7.9254 10.0000 45 15.8346 1.1076 10.0000 17.0000 7.8427 10.0000 46 16.1741 1.0562 10.0000 17.4097 7.7465 10.0000 47 16.5119 1.0000 10.0000 17.8157 7.6357 10.0000 48 16.8497 0.9384 10.0000 18.2178 7.5119 10.0000 49 17.1859 0.8724 10.0000 18.6151 7.3741 10.0000 50 17.5216 0.8011 10.0000 19.0081 7.2227 10.0000 51 17.8557 0.7249 10.0000 19.3951 7.0584 10.0000 52 18.1886 0.6432 10.0000 19.7768 6.8811 10.0000 53 18.5205 0.5568 10.0000 20.1524 6.6908 10.0000 54 18.8508 0.4643 10.0000 20.5216 6.4886 10.0000 55 19.1800 0.3670 10.0000 20.8838 6.2746 10.0000 56 19.5070 0.2649 10.0000 21.2395 6.0492 10.0000 57 19.8324 0.1568 10.0000 21.5876 5.8130 10.0000 58 20.1562 0.0432 10.0000 21.9281 5.5659 10.0000 59 20.4778 −0.0751 10.0000 22.2611 5.3086 10.0000 60 20.7978 −0.1995 10.0000 22.5865 5.0416 10.0000 61 21.1151 −0.3292 10.0000 22.9038 4.7654 10.0000 62 21.4303 −0.4638 10.0000 23.2135 4.4805 10.0000 63 21.7432 −0.6043 10.0000 23.5162 4.1881 10.0000 64 22.0535 −0.7503 10.0000 23.8124 3.8892 10.0000 65 22.3616 −0.9011 10.0000 24.1016 3.5832 10.0000 66 22.6670 −1.0573 10.0000 24.3843 3.2714 10.0000 67 22.9692 −1.2195 10.0000 24.6605 2.9541 10.0000 68 23.2686 −1.3865 10.0000 24.9303 2.6314 10.0000 69 23.5654 −1.5584 10.0000 25.1946 2.3032 10.0000 70 23.8584 −1.7362 10.0000 25.4530 1.9714 10.0000 71 24.1486 −1.9189 10.0000 25.7070 1.6357 10.0000 72 24.4357 −2.1065 10.0000 25.9562 1.2968 10.0000 73 24.7195 −2.2989 10.0000 26.2016 0.9551 10.0000 74 25.0005 −2.4962 10.0000 26.4432 0.6103 10.0000 75 25.2778 −2.6973 10.0000 26.6800 0.2627 10.0000 76 25.5519 −2.9032 10.0000 26.9130 −0.0881 10.0000 77 25.8227 −3.1141 10.0000 27.1416 −0.4416 10.0000 78 26.0903 −3.3286 10.0000 27.3659 −0.7973 10.0000 79 26.3535 −3.5481 10.0000 27.5865 −1.1557 10.0000 80 26.6130 −3.7724 10.0000 27.8038 −1.5162 10.0000 81 26.8681 −4.0016 10.0000 28.0184 −1.8784 10.0000 82 27.1184 −4.2362 10.0000 28.2308 −2.2416 10.0000 83 27.3643 −4.4751 10.0000 28.4416 −2.6059 10.0000 84 27.6049 −4.7195 10.0000 28.6508 −2.9708 10.0000 85 27.8405 −4.9686 10.0000 28.8584 −3.3368 10.0000 86 28.0719 −5.2222 10.0000 29.0649 −3.7038 10.0000 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27.0054 −9.0000 70.0000 27.9005 −8.5822 70.0000 96 27.1638 −9.2595 70.0000 28.0514 −8.8989 70.0000 97 27.3222 −9.5195 70.0000 28.2000 −9.2168 70.0000 98 27.4897 −9.7724 70.0000 28.2481 −9.5573 70.0000 99 27.7562 −9.9054 70.0000 28.0465 −9.8341 70.0000 100 28.0465 −9.8341 70.0000 5.1946 6.8195 80.0000 1 5.1946 6.8195 80.0000 5.1827 7.1551 80.0000 2 5.3562 6.5730 80.0000 5.3259 7.4627 80.0000 3 5.5930 6.3951 80.0000 5.5449 7.7238 80.0000 4 5.8611 6.2676 80.0000 5.8000 7.9497 80.0000 5 6.1427 6.1730 80.0000 6.0762 8.1497 80.0000 6 6.4319 6.1054 80.0000 6.3670 8.3281 80.0000 7 6.7259 6.0611 80.0000 6.6676 8.4886 80.0000 8 7.0216 6.0346 80.0000 6.9762 8.6341 80.0000 9 7.3184 6.0195 80.0000 7.2903 8.7665 80.0000 10 7.6151 6.0097 80.0000 7.6092 8.8865 80.0000 11 7.9124 6.0005 80.0000 7.9324 8.9957 80.0000 12 8.2092 5.9892 80.0000 8.2595 9.0935 80.0000 13 8.5059 5.9730 80.0000 8.5892 9.1800 80.0000 14 8.8022 5.9519 80.0000 8.9216 9.2551 80.0000 15 9.0984 5.9249 80.0000 9.2568 9.3184 80.0000 16 9.3935 5.8930 80.0000 9.5941 9.3692 80.0000 17 9.6886 5.8557 80.0000 9.9330 9.4070 80.0000 18 9.9827 5.8141 80.0000 10.2730 9.4319 80.0000 19 10.2762 5.7676 80.0000 10.6141 9.4438 80.0000 20 10.5686 5.7157 80.0000 10.9551 9.4422 80.0000 21 10.8605 5.6584 80.0000 11.2957 9.4270 80.0000 22 11.1508 5.5951 80.0000 11.6357 9.3989 80.0000 23 11.4395 5.5254 80.0000 11.9741 9.3573 80.0000 24 11.7270 5.4497 80.0000 12.3108 9.3027 80.0000 25 12.0124 5.3676 80.0000 12.6449 9.2351 80.0000 26 12.2962 5.2795 80.0000 12.9762 9.1551 80.0000 27 12.5778 5.1854 80.0000 13.3049 9.0622 80.0000 28 12.8578 5.0854 80.0000 13.6292 8.9578 80.0000 29 13.1351 4.9789 80.0000 13.9497 8.8416 80.0000 30 13.4108 4.8670 80.0000 14.2659 8.7141 80.0000 31 13.6832 4.7492 80.0000 14.5778 8.5757 80.0000 32 13.9530 4.6249 80.0000 14.8849 8.4276 80.0000 33 14.2205 4.4951 80.0000 15.1870 8.2692 80.0000 34 14.4849 4.3595 80.0000 15.4838 8.1016 80.0000 35 14.7465 4.2184 80.0000 15.7757 7.9249 80.0000 36 15.0049 4.0719 80.0000 16.0622 7.7400 80.0000 37 15.2605 3.9205 80.0000 16.3432 7.5470 80.0000 38 15.5135 3.7643 80.0000 16.6189 7.3459 80.0000 39 15.7632 3.6032 80.0000 16.8892 7.1378 80.0000 40 16.0097 3.4378 80.0000 17.1541 6.9232 80.0000 41 16.2535 3.2676 80.0000 17.4141 6.7027 80.0000 42 16.4946 3.0935 80.0000 17.6686 6.4757 80.0000 43 16.7324 2.9157 80.0000 17.9189 6.2438 80.0000 44 16.9670 2.7335 80.0000 18.1643 6.0070 80.0000 45 17.1989 2.5481 80.0000 18.4049 5.7654 80.0000 46 17.4286 2.3589 80.0000 18.6405 5.5189 80.0000 47 17.6551 2.1665 80.0000 18.8724 5.2686 80.0000 48 17.8789 1.9714 80.0000 19.0995 5.0141 80.0000 49 18.1000 1.7730 80.0000 19.3227 4.7568 80.0000 50 18.3189 1.5719 80.0000 19.5422 4.4957 80.0000 51 18.5351 1.3681 80.0000 19.7584 4.2319 80.0000 52 18.7492 1.1616 80.0000 19.9714 3.9654 80.0000 53 18.9605 0.9530 80.0000 20.1811 3.6962 80.0000 54 19.1697 0.7422 80.0000 20.3876 3.4249 80.0000 55 19.3768 0.5292 80.0000 20.5914 3.1514 80.0000 56 19.5816 0.3135 80.0000 20.7924 2.8762 80.0000 57 19.7843 0.0968 80.0000 20.9908 2.5984 80.0000 58 19.9854 −0.1222 80.0000 21.1859 2.3189 80.0000 59 20.1843 −0.3432 80.0000 21.3795 2.0378 80.0000 60 20.3816 −0.5654 80.0000 21.5703 1.7551 80.0000 61 20.5768 −0.7892 80.0000 21.7589 1.4714 80.0000 62 20.7708 −1.0146 80.0000 21.9454 1.1859 80.0000 63 20.9627 −1.2416 80.0000 22.1303 0.8989 80.0000 64 21.1530 −1.4697 80.0000 22.3130 0.6114 80.0000 65 21.3416 −1.6995 80.0000 22.4941 0.3222 80.0000 66 21.5286 −1.9303 80.0000 22.6735 0.0324 80.0000 67 21.7141 −2.1622 80.0000 22.8514 −0.2584 80.0000 68 21.8984 −2.3951 80.0000 23.0281 −0.5508 80.0000 69 22.0811 −2.6297 80.0000 23.2027 −0.8432 80.0000 70 22.2627 −2.8649 80.0000 23.3751 −1.1373 80.0000 71 22.4427 −3.1011 80.0000 23.5465 −1.4324 80.0000 72 22.6222 −3.3384 80.0000 23.7157 −1.7286 80.0000 73 22.8000 −3.5762 80.0000 23.8827 −2.0259 80.0000 74 22.9773 −3.8146 80.0000 24.0481 −2.3243 80.0000 75 23.1535 −4.0541 80.0000 24.2119 −2.6232 80.0000 76 23.3286 −4.2941 80.0000 24.3741 −2.9232 80.0000 77 23.5032 −4.5346 80.0000 24.5346 −3.2243 80.0000 78 23.6762 −4.7762 80.0000 24.6941 −3.5259 80.0000 79 23.8481 −5.0184 80.0000 24.8519 −3.8281 80.0000 80 24.0189 −5.2616 80.0000 25.0092 −4.1308 80.0000 81 24.1881 −5.5059 80.0000 25.1659 −4.4335 80.0000 82 24.3551 −5.7514 80.0000 25.3227 −4.7362 80.0000 83 24.5205 −5.9984 80.0000 25.4789 −5.0395 80.0000 84 24.6838 −6.2470 80.0000 25.6351 −5.3427 80.0000 85 24.8449 −6.4962 80.0000 25.7914 −5.6459 80.0000 86 25.0043 −6.7470 80.0000 25.9476 −5.9492 80.0000 87 25.1616 −6.9989 80.0000 26.1038 −6.2524 80.0000 88 25.3184 −7.2514 80.0000 26.2600 −6.5551 80.0000 89 25.4741 −7.5049 80.0000 26.4157 −6.8589 80.0000 90 25.6292 −7.7584 80.0000 26.5708 −7.1627 80.0000 91 25.7832 −8.0119 80.0000 26.7254 −7.4665 80.0000 92 25.9378 −8.2659 80.0000 26.8784 −7.7714 80.0000 93 26.0914 −8.5205 80.0000 27.0297 −8.0768 80.0000 94 26.2454 −8.7746 80.0000 27.1795 −8.3832 80.0000 95 26.3989 −9.0286 80.0000 27.3276 −8.6908 80.0000 96 26.5530 −9.2832 80.0000 27.4735 −8.9989 80.0000 97 26.7065 −9.5373 80.0000 27.6162 −9.3086 80.0000 98 26.8649 −9.7886 80.0000 27.6259 −9.6416 80.0000 99 27.1162 −9.9368 80.0000 27.4043 −9.8908 80.0000 100 27.4043 −9.8908 80.0000 5.6151 7.7384 90.0000 1 5.6151 7.7384 90.0000 5.6362 8.0627 90.0000 2 5.7476 7.4811 90.0000 5.8022 8.3459 90.0000 3 5.9616 7.2838 90.0000 6.0357 8.5773 90.0000 4 6.2124 7.1346 90.0000 6.3022 8.7714 90.0000 5 6.4795 7.0178 90.0000 6.5865 8.9378 90.0000 6 6.7568 6.9270 90.0000 6.8832 9.0816 90.0000 7 7.0411 6.8595 90.0000 7.1881 9.2070 90.0000 8 7.3286 6.8103 90.0000 7.4995 9.3157 90.0000 9 7.6178 6.7703 90.0000 7.8151 9.4114 90.0000 10 7.9076 6.7346 90.0000 8.1341 9.4935 90.0000 11 8.1968 6.6984 90.0000 8.4562 9.5638 90.0000 12 8.4859 6.6584 90.0000 8.7811 9.6216 90.0000 13 8.7746 6.6146 90.0000 9.1076 9.6676 90.0000 14 9.0627 6.5659 90.0000 9.4351 9.7011 90.0000 15 9.3497 6.5119 90.0000 9.7643 9.7216 90.0000 16 9.6357 6.4535 90.0000 10.0941 9.7292 90.0000 17 9.9205 6.3908 90.0000 10.4238 9.7232 90.0000 18 10.2049 6.3238 90.0000 10.7530 9.7049 90.0000 19 10.4876 6.2524 90.0000 11.0811 9.6730 90.0000 20 10.7692 6.1757 90.0000 11.4076 9.6281 90.0000 21 11.0497 6.0941 90.0000 11.7324 9.5714 90.0000 22 11.3281 6.0070 90.0000 12.0546 9.5016 90.0000 23 11.6049 5.9141 90.0000 12.3741 9.4200 90.0000 24 11.8795 5.8157 90.0000 12.6903 9.3265 90.0000 25 12.1524 5.7114 90.0000 13.0027 9.2222 90.0000 26 12.4232 5.6022 90.0000 13.3114 9.1065 90.0000 27 12.6914 5.4870 90.0000 13.6162 8.9805 90.0000 28 12.9573 5.3659 90.0000 13.9162 8.8443 90.0000 29 13.2205 5.2400 90.0000 14.2119 8.6984 90.0000 30 13.4811 5.1086 90.0000 14.5032 8.5438 90.0000 31 13.7389 4.9719 90.0000 14.7892 8.3800 90.0000 32 13.9935 4.8297 90.0000 15.0708 8.2081 90.0000 33 14.2454 4.6822 90.0000 15.3470 8.0286 90.0000 34 14.4946 4.5297 90.0000 15.6184 7.8416 90.0000 35 14.7405 4.3724 90.0000 15.8854 7.6476 90.0000 36 14.9832 4.2103 90.0000 16.1465 7.4470 90.0000 37 15.2232 4.0438 90.0000 16.4032 7.2400 90.0000 38 15.4595 3.8730 90.0000 16.6551 7.0270 90.0000 39 15.6935 3.6984 90.0000 16.9022 6.8086 90.0000 40 15.9243 3.5195 90.0000 17.1443 6.5849 90.0000 41 16.1524 3.3373 90.0000 17.3816 6.3562 90.0000 42 16.3773 3.1514 90.0000 17.6151 6.1232 90.0000 43 16.5995 2.9622 90.0000 17.8438 5.8859 90.0000 44 16.8189 2.7697 90.0000 18.0686 5.6443 90.0000 45 17.0357 2.5741 90.0000 18.2892 5.3995 90.0000 46 17.2497 2.3757 90.0000 18.5065 5.1514 90.0000 47 17.4611 2.1746 90.0000 18.7200 4.9005 90.0000 48 17.6703 1.9708 90.0000 18.9303 4.6465 90.0000 49 17.8768 1.7643 90.0000 19.1373 4.3903 90.0000 50 18.0811 1.5557 90.0000 19.3416 4.1314 90.0000 51 18.2832 1.3454 90.0000 19.5427 3.8697 90.0000 52 18.4832 1.1330 90.0000 19.7411 3.6065 90.0000 53 18.6816 0.9184 90.0000 19.9368 3.3411 90.0000 54 18.8778 0.7027 90.0000 20.1297 3.0735 90.0000 55 19.0724 0.4849 90.0000 20.3200 2.8049 90.0000 56 19.2649 0.2654 90.0000 20.5081 2.5341 90.0000 57 19.4557 0.0443 90.0000 20.6941 2.2616 90.0000 58 19.6449 −0.1778 90.0000 20.8778 1.9876 90.0000 59 19.8324 −0.4016 90.0000 21.0595 1.7124 90.0000 60 20.0178 −0.6270 90.0000 21.2389 1.4362 90.0000 61 20.2016 −0.8535 90.0000 21.4168 1.1584 90.0000 62 20.3843 −1.0816 90.0000 21.5924 0.8795 90.0000 63 20.5649 −1.3108 90.0000 21.7665 0.5995 90.0000 64 20.7449 −1.5411 90.0000 21.9384 0.3184 90.0000 65 20.9227 −1.7719 90.0000 22.1092 0.0362 90.0000 66 21.1000 −2.0038 90.0000 22.2778 −0.2470 90.0000 67 21.2762 −2.2368 90.0000 22.4454 −0.5308 90.0000 68 21.4514 −2.4703 90.0000 22.6114 −0.8162 90.0000 69 21.6254 −2.7049 90.0000 22.7757 −1.1016 90.0000 70 21.7984 −2.9400 90.0000 22.9384 −1.3886 90.0000 71 21.9708 −3.1751 90.0000 23.1000 −1.6757 90.0000 72 22.1422 −3.4114 90.0000 23.2600 −1.9643 90.0000 73 22.3130 −3.6481 90.0000 23.4184 −2.2535 90.0000 74 22.4827 −3.8859 90.0000 23.5757 −2.5432 90.0000 75 22.6519 −4.1238 90.0000 23.7314 −2.8335 90.0000 76 22.8195 −4.3627 90.0000 23.8859 −3.1249 90.0000 77 22.9865 −4.6022 90.0000 24.0389 −3.4168 90.0000 78 23.1519 −4.8427 90.0000 24.1908 −3.7097 90.0000 79 23.3168 −5.0838 90.0000 24.3411 −4.0027 90.0000 80 23.4800 −5.3254 90.0000 24.4908 −4.2968 90.0000 81 23.6422 −5.5686 90.0000 24.6400 −4.5908 90.0000 82 23.8027 −5.8119 90.0000 24.7881 −4.8854 90.0000 83 23.9616 −6.0568 90.0000 24.9368 −5.1800 90.0000 84 24.1195 −6.3022 90.0000 25.0849 −5.4741 90.0000 85 24.2762 −6.5486 90.0000 25.2335 −5.7686 90.0000 86 24.4314 −6.7962 90.0000 25.3827 −6.0627 90.0000 87 24.5849 −7.0443 90.0000 25.5319 −6.3562 90.0000 88 24.7373 −7.2930 90.0000 25.6816 −6.6503 90.0000 89 24.8892 −7.5427 90.0000 25.8314 −6.9438 90.0000 90 25.0395 −7.7924 90.0000 25.9805 −7.2378 90.0000 91 25.1897 −8.0432 90.0000 26.1297 −7.5319 90.0000 92 25.3389 −8.2941 90.0000 26.2773 −7.8270 90.0000 93 25.4881 −8.5449 90.0000 26.4238 −8.1222 90.0000 94 25.6378 −8.7957 90.0000 26.5681 −8.4184 90.0000 95 25.7870 −9.0465 90.0000 26.7108 −8.7157 90.0000 96 25.9368 −9.2968 90.0000 26.8514 −9.0141 90.0000 97 26.0876 −9.5470 90.0000 26.9881 −9.3141 90.0000 98 26.2459 −9.7914 90.0000 26.9914 −9.6368 90.0000 99 26.4962 −9.9303 90.0000 26.7784 −9.8795 90.0000 100 26.7784 −9.8795 90.0000 6.0595 8.6341 ####### 1 6.0595 8.6341 100.0000 6.0427 8.9486 ####### 2 6.2108 8.3924 100.0000 6.1946 9.2276 ####### 3 6.4259 8.2038 100.0000 6.4286 9.4454 ####### 4 6.6697 8.0541 100.0000 6.6989 9.6168 ####### 5 6.9303 7.9341 100.0000 6.9870 9.7557 ####### 6 7.2005 7.8384 100.0000 7.2865 9.8692 ####### 7 7.4768 7.7611 100.0000 7.5935 9.9616 ####### 8 7.7557 7.6968 100.0000 7.9049 10.0357 ####### 9 8.0368 7.6378 100.0000 8.2200 10.0935 ####### 10 8.3173 7.5789 100.0000 8.5373 10.1368 ####### 11 8.5968 7.5157 100.0000 8.8562 10.1659 ####### 12 8.8751 7.4476 100.0000 9.1757 10.1822 ####### 13 9.1524 7.3735 100.0000 9.4962 10.1849 ####### 14 9.4281 7.2951 100.0000 9.8162 10.1746 ####### 15 9.7027 7.2130 100.0000 10.1357 10.1508 ####### 16 9.9762 7.1259 100.0000 10.4541 10.1146 ####### 17 10.2481 7.0346 100.0000 10.7703 10.0659 ####### 18 10.5178 6.9384 100.0000 11.0849 10.0049 ####### 19 10.7859 6.8373 100.0000 11.3968 9.9324 ####### 20 11.0519 6.7303 100.0000 11.7059 9.8492 ####### 21 11.3162 6.6184 100.0000 12.0119 9.7541 ####### 22 11.5778 6.5011 100.0000 12.3141 9.6481 ####### 23 11.8373 6.3795 100.0000 12.6119 9.5308 ####### 24 12.0946 6.2530 100.0000 12.9059 9.4032 ####### 25 12.3492 6.1211 100.0000 13.1957 9.2670 ####### 26 12.6016 5.9843 100.0000 13.4816 9.1222 ####### 27 12.8508 5.8427 100.0000 13.7622 8.9686 ####### 28 13.0973 5.6962 100.0000 14.0389 8.8070 ####### 29 13.3411 5.5454 100.0000 14.3108 8.6373 ####### 30 13.5816 5.3897 100.0000 14.5778 8.4605 ####### 31 13.8195 5.2297 100.0000 14.8405 8.2773 ####### 32 14.0546 5.0654 100.0000 15.0984 8.0870 ####### 33 14.2865 4.8968 100.0000 15.3514 7.8908 ####### 34 14.5157 4.7243 100.0000 15.5995 7.6886 ####### 35 14.7416 4.5481 100.0000 15.8438 7.4811 ####### 36 14.9649 4.3681 100.0000 16.0832 7.2686 ####### 37 15.1854 4.1849 100.0000 16.3184 7.0508 ####### 38 15.4027 3.9978 100.0000 16.5492 6.8292 ####### 39 15.6178 3.8081 100.0000 16.7757 6.6027 ####### 40 15.8303 3.6157 100.0000 16.9984 6.3724 ####### 41 16.0400 3.4200 100.0000 17.2173 6.1389 ####### 42 16.2470 3.2216 100.0000 17.4324 5.9011 ####### 43 16.4519 3.0211 100.0000 17.6438 5.6611 ####### 44 16.6541 2.8184 100.0000 17.8519 5.4173 ####### 45 16.8546 2.6130 100.0000 18.0568 5.1714 ####### 46 17.0524 2.4059 100.0000 18.2584 4.9222 ####### 47 17.2486 2.1962 100.0000 18.4573 4.6714 ####### 48 17.4427 1.9854 100.0000 18.6530 4.4178 ####### 49 17.6346 1.7724 100.0000 18.8465 4.1622 ####### 50 17.8249 1.5578 100.0000 19.0368 3.9049 ####### 51 18.0130 1.3416 100.0000 19.2249 3.6454 ####### 52 18.1995 1.1238 100.0000 19.4108 3.3843 ####### 53 18.3843 0.9043 100.0000 19.5941 3.1222 ####### 54 18.5670 0.6838 100.0000 19.7757 2.8578 ####### 55 18.7486 0.4616 100.0000 19.9546 2.5924 ####### 56 18.9286 0.2384 100.0000 20.1324 2.3259 ####### 57 19.1070 0.0141 100.0000 20.3076 2.0578 ####### 58 19.2843 −0.2114 100.0000 20.4816 1.7892 ####### 59 19.4605 −0.4378 100.0000 20.6535 1.5189 ####### 60 19.6351 −0.6649 100.0000 20.8238 1.2476 ####### 61 19.8086 −0.8930 100.0000 20.9924 0.9751 ####### 62 19.9811 −1.1222 100.0000 21.1595 0.7016 ####### 63 20.1524 −1.3519 100.0000 21.3249 0.4276 ####### 64 20.3232 −1.5827 100.0000 21.4886 0.1524 ####### 65 20.4924 −1.8141 100.0000 21.6514 −0.1232 ####### 66 20.6611 −2.0459 100.0000 21.8124 −0.4000 ####### 67 20.8286 −2.2784 100.0000 21.9724 −0.6778 ####### 68 20.9957 −2.5119 100.0000 22.1303 −0.9562 ####### 69 21.1616 −2.7454 100.0000 22.2870 −1.2357 ####### 70 21.3265 −2.9800 100.0000 22.4427 −1.5157 ####### 71 21.4908 −3.2146 100.0000 22.5968 −1.7968 ####### 72 21.6546 −3.4503 100.0000 22.7492 −2.0784 ####### 73 21.8173 −3.6859 100.0000 22.9005 −2.3605 ####### 74 21.9795 −3.9227 100.0000 23.0503 −2.6438 ####### 75 22.1411 −4.1595 100.0000 23.1984 −2.9276 ####### 76 22.3016 −4.3973 100.0000 23.3454 −3.2119 ####### 77 22.4616 −4.6351 100.0000 23.4914 −3.4973 ####### 78 22.6205 −4.8735 100.0000 23.6362 −3.7827 ####### 79 22.7789 −5.1130 100.0000 23.7805 −4.0686 ####### 80 22.9362 −5.3524 100.0000 23.9243 −4.3551 ####### 81 23.0924 −5.5930 100.0000 24.0676 −4.6416 ####### 82 23.2476 −5.8341 100.0000 24.2108 −4.9281 ####### 83 23.4022 −6.0757 100.0000 24.3541 −5.2146 ####### 84 23.5551 −6.3178 100.0000 24.4978 −5.5005 ####### 85 23.7070 −6.5611 100.0000 24.6422 −5.7865 ####### 86 23.8578 −6.8049 100.0000 24.7865 −6.0724 ####### 87 24.0076 −7.0497 100.0000 24.9314 −6.3584 ####### 88 24.1557 −7.2951 100.0000 25.0762 −6.6438 ####### 89 24.3027 −7.5411 100.0000 25.2205 −6.9297 ####### 90 24.4486 −7.7881 100.0000 25.3649 −7.2157 ####### 91 24.5946 −8.0351 100.0000 25.5086 −7.5022 ####### 92 24.7395 −8.2822 100.0000 25.6514 −7.7886 ####### 93 24.8849 −8.5297 100.0000 25.7924 −8.0762 ####### 94 25.0297 −8.7768 100.0000 25.9319 −8.3643 ####### 95 25.1757 −9.0238 100.0000 26.0697 −8.6535 ####### 96 25.3222 −9.2703 100.0000 26.2059 −8.9438 ####### 97 25.4697 −9.5162 100.0000 26.3395 −9.2346 ####### 98 25.6324 −9.7508 100.0000 26.3551 −9.5486 ####### 99 25.8886 −9.8659 100.0000 26.1622 −9.7957 ####### 100 26.1622 −9.7957 100.0000

It will also be appreciated that the airfoil 52 disclosed in the above Table I may be scaled up or down geometrically for use in other similar turbine designs. Consequently, the coordinate values set forth in Table I may be scaled upwardly or downwardly such that the airfoil profile shape remains unchanged. A scaled version of the coordinates in Table I would be represented by X, Y and Z coordinate values of Table I, with the X, Y and Z non-dimensional coordinate values converted to inches, multiplied or divided by a constant number.

An important term in this disclosure is “profile”. The profile is the range of the variation between measured points on an airfoil surface and the ideal position listed in Table I. The actual profile on a manufactured turbine rotor blade will be different than those in Table I and the design is robust to this variation meaning that mechanical and aerodynamic function are not impaired. As noted above, a + or −5% profile tolerance is used herein. The X, Y and Z values are all non-dimensionalized relative to the airfoil height.

The disclosed airfoil shape optimizes and is specific to the machine conditions and specifications. It provides a unique profile to achieve 1) interaction between other stages in the turbine 10; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical blade loadings. The disclosed loci of points defined in Table I allow the gas turbine 12 or any other suitable turbine to run in an efficient, safe and smooth manner. As also noted, any scale of the disclosed airfoil 52 may be adopted as long as 1) interaction between other stages in the pressure turbine 10; 2) aerodynamic efficiency; and 3) normalized aerodynamic and mechanical blade loadings are maintained in the scaled turbine.

This written description uses examples to disclose the invention, including the best mode, and also to enable any person skilled in the art to practice the invention, including making and using any devices or systems and performing any incorporated methods. The patentable scope of the invention is defined by the claims, and may include other examples that occur to those skilled in the art. Such other examples are intended to be within the scope of the claims if they have structural elements that do not differ from the literal language of the claims, or if they include equivalent structural elements with insubstantial differences from the literal languages of the claims. 

What is claimed is:
 1. A turbine rotor blade including an airfoil having an airfoil shape, the airfoil having a nominal profile substantially in accordance with Cartesian coordinate values of X, Y and Z set forth in Table I, wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each distance Z, the airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape.
 2. The turbine rotor blade as in claim 1, forming part of a stage of a turbine.
 3. The turbine rotor blade as in claim 1, wherein the turbine rotor blade forms a part of a third stage of turbine rotor blades of a turbine of a gas turbine.
 4. The turbine rotor blade as in claim 1, wherein the airfoil shape lies in an envelope within +/−5% in a direction normal to any airfoil surface location.
 5. The turbine rotor blade as in claim 1, wherein a height of the turbine rotor blade is 10 inches to 40 inches.
 6. A turbine rotor blade including an airfoil having a suction-side uncoated nominal airfoil profile substantially in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table I, wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances in inches by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil in inches, and wherein X and Y are distances in inches which, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at the Z distances being joined smoothly with one another to form a complete suction-side airfoil shape, the X, Y and Z distances being scalable as a function of the same constant or number to provide a scaled-up or scaled-down airfoil.
 7. The turbine rotor blade as in claim 6, forming part of a stage of a turbine.
 8. The turbine rotor blade as in claim 6, wherein the turbine rotor blade forms a part of a third stage of turbine rotor blades of a turbine of a gas turbine.
 9. The turbine rotor blade as in claim 6, wherein the suction-side airfoil shape lies in an envelope within +/−5% in a direction normal to any suction-side airfoil surface location.
 10. The turbine rotor blade as in claim 6, wherein a height of the turbine rotor blade is about 10 inches to about 40 inches.
 11. A turbine comprising a turbine wheel having a plurality of turbine rotor blades, each of the turbine rotor blades including an airfoil having a suction-side airfoil shape, the airfoil having a nominal profile substantially in accordance with suction-side Cartesian coordinate values of X, Y and Z set forth in Table I wherein the Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the Cartesian coordinate values of X, Y and Z by a height of the airfoil, and wherein X and Y are distances which, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at the Z distances being joined smoothly with one another to form a complete suction-side airfoil shape.
 12. The turbine as in claim 11, wherein the turbine wheel comprises a stage of the turbine.
 13. The turbine as in claim 11, wherein X represents a distance parallel to the turbine axis of rotation.
 14. The turbine as in claim 11, wherein the suction-side airfoil shape lies in an envelope within +/−5% in a direction normal to any suction-side airfoil surface location.
 15. The turbine as in claim 11, wherein a height of the turbine rotor blade is about 10 inches to about 40 inches.
 16. The turbine as in claim 11, wherein each of the turbine rotor blades includes an airfoil having a pressure-side airfoil shape, the airfoil having a nominal profile substantially in accordance with pressure-side Cartesian coordinate values of X, Y and Z set forth in Table I wherein the pressure-side Cartesian coordinate values of X, Y and Z are non-dimensional values from 0% to 100% convertible to dimensional distances by multiplying the pressure-side Cartesian coordinate values of X, Y and Z by a height of the airfoil, and wherein X and Y are distances which, when connected by smooth continuing arcs, define airfoil profile sections at each Z distance, the airfoil profile sections at Z distances being joined smoothly with one another to form a complete airfoil shape.
 17. The turbine as in claim 16, wherein the turbine rotor blade is a third stage turbine rotor blade.
 18. The turbine as in claim 14, wherein X represents a distance parallel to the turbine axis of rotation.
 19. The turbine as in claim 16, wherein the airfoil shape lies in an envelope within +/−5% in a direction normal to any airfoil surface location.
 20. The turbine as in claim 16, wherein a height of the turbine rotor blade is about 10 inches to about 40 inches. 